where \(x^{AA^{\prime }}\)  denotes any point in \(\gamma \) , and
the factor \(i\)  is here introduced only to follow standard twistor conventions.
The entire light ray \(\gamma \)  is then in principle represented by the pair of spinors \((\lambda _{A},\mu ^{A^{\prime }})\) .
However, the same \(\gamma \)  is obtained if we multiply \(\lambda _{A}\)  and \(\mu ^{A^{\prime }}\)  by a non-zero
complex number. Thus, \(\gamma \)  can be represented as a point in \((\mathbb {C}^{2*}\oplus \bar{\mathbb {C}}^{2})/\sim \) ,
where \(\sim \)  refers to the equivalence relation just mentioned. The space
\(\mathbb {PT}:=(\mathbb {C}^{2*}\oplus \bar{\mathbb {C}}^{2})/\sim \)  is called (projective) twistor space
(we see that it is isomorphic to the complex projective 3-space \(\mathbb {CP}^{3}\)  Twistor space
is actually \(\mathbb {CP}^{3}\backslash \mathbb {CP}^{1}\)  (see e.g. [1]}, [2]}),
but we do not need to make this distinction for our elementary presentation.).
Actually, real light rays lie in a submanifold of \(\mathbb {PT}\) , since for real \(x^{AA^{\prime }}\)  one has the constraint
\(\mu ^{A^{\prime }}\bar{\lambda }_{A^{\prime }}+\bar{\mu }^{A}\lambda _{A}=0\) , as follows from (REF ).
On the other hand, given a point \((\lambda _{A},\mu ^{A^{\prime }})\)  in twistor space \(\mathbb {PT}\) , what does this correspond to in spacetime?
The answer is the set of \(x^{AA^{\prime }}\)  such that (REF ) holds.
If \(\mu ^{A^{\prime }}\bar{\lambda }_{A^{\prime }}+\bar{\mu }^{A}\lambda _{A}\ne 0\) , (REF ) has no
real solutions for \(x^{AA^{\prime }}\) , so one needs to complexify spacetime in order to have solutions.
In the complexified spacetime \(\mathbb {CM}\) , suppose that \(y^{AA^{\prime }}\)  is a solution to (REF ),
i.e. \(y^{AA^{\prime }}\lambda _{A}=i\mu ^{A^{\prime }}\) , then we see that \(x^{AA^{\prime }}=\lambda ^{A}\pi ^{A^{\prime }}+y^{AA^{\prime }}\)  is also a solution
for any spinor \(\pi ^{A^{\prime }}\) .
Thus the set of points in \(\mathbb {CM}\)  satisfying (REF ) is a 2-dimensional (complex) plane \(\Sigma \) .
For any two points \(x,y\)  in \(\Sigma \subset \mathbb {CM}\)  we have
\(x^{AA^{\prime }}-y^{AA^{\prime }}=\lambda ^{A}\pi ^{A^{\prime }}\)  for some \(\pi ^{A^{\prime }}\) , thus vectors tangents to \(\Sigma \) 
are of the form \(\lambda ^{A}\pi ^{A^{\prime }}\)  where \(\lambda ^{A}\)  is fixed and \(\pi ^{A^{\prime }}\)  varies.
This implies that any tangent to \(\Sigma \)  is null, and likewise the inner product
between any two tangent vectors vanishes. In other words,
\(\eta |_{T\Sigma } \equiv 0\) 
